Quadrature-quadrature phase shift keying with constant envelope

ABSTRACT

Methods for modulating and demodulating digital data streams utilize a quadrature-quadrature phase shift keying data transmission arrangement to acheive a 100% increase in the bandwidth efficiency over known systems, such as minimum shift keying. Known arrangements utilize two dimensional data transmission. However, Q 2  PSK, in accordance with the invention, provides four dimensional transmission which doubles the rate of data transmission for a given bandwidth, at the expense of approximately 45% increase in the average energy per bit. The input data stream is demultiplexed to form four demultiplexed data streams which are formed of demultiplexed data bits, each such stream being coded to form a stream of data words formed of a predetermined number of data pulses and a parity check bit. Each such data stream is combined with a signal having carrier and data pulse-shaping components. Additionally, the data pulse-shaping components have a quadrature phase relationship with each other, thereby adding the additional two dimensions of data transmission capacity within a constant envelope.

This is a continuation-in-part, of application Ser. No. 640,057 filedAug. 13, 1984, now U.S. Pat. No. 4,680,777.

BACKGROUND OF THE INVENTION

This invention relates generally to transmission and modulation systemsfor digital data, and more particularly, to a digital data modulationsystem wherein bandwidth efficiency and the data transmission rate areimproved by demultiplexing the digital data into a plurality of datachannels, wherein the data pulses are shaped by signals havingpreselected characterizing functions, illustratively sinusoidal andcosinusoidal functions, which have a quadrature phase relationship, andwherein the transmitted signal has a constant envelope.

Spectral congestion due to ever increasing demand for digitaltransmission calls for spectrally efficient modulation schemes.Spectrally efficient modulation can be loosely said to be the use ofpower to save bandwidth, much as coding may be said to be the use ofbandwidth to save power. In other words the primary objective of aspectrally efficient modulation scheme is to maximize the bandwidthefficiency (b), defined as the ratio of data rate (R_(b)) to channelbandwidth (W). Since a signal can not be both strictly duration-limitedand strictly band-limited, there are two approaches in designing aspectrally efficient data transmission scheme. One is the band-limitingapproach; the other is the time-limiting approach. In the former, astrictly bandlimited spectral shape is carefully chosen for the datapulse so as to satisfy the Nyquist criterion of zero intersymbolinterference (ISI). In the latter, the data pulse is designed to have ashort duration and the definition of bandwidth is somewhat relativedepending on the situation involved. The latter approach is followedherein.

Like bandwidth, power is also a costly resource in radio transmission.So another objective in designing a high rate data transmission schemeis to reduce the average energy per bit (E_(b)) for achieving aspecified bit error rate (BER). The bit error rate performance of twoschemes are usually compared under the assumption of an ideal channelcorrupted only by additive white Gaussian noise (AWGN). Suppose the twosided power spectral density of noise is N₀ /2. Then a standardparameter for comparing two modulation schemes is the energy efficiency(e); it is the ratio E_(b) /N₀ required to achieve a BER of 10⁻⁵.

The energy efficiency mostly depends on the signal space geometry. Thebandwidth efficiency depends on two factors; firstly the basic waveformsof the data shaping pulses and secondly the utilization of all possiblesignal dimensions available within the given transmission bandwidth. Indata communication, the notion of increasing the rate of transmission byincreasing the number of dimensions became prominent when peopleswitched from binary phase shift keying (BPSK) to quadrature phase shiftkeying (QPSK). Modulation studies during the last fifteen years proposedseveral modifications of QPSK. Of these, offset quadrature phase shiftkeying (OQPSK) and minimum shift keying (MSK) have gained popularitybecause of their several attributes. So any new spectrally efficientscheme ought to be tested in the light of the spectral efficiencies ofthese two.

BPSK is an antipodal signalling scheme; it uses two opposite phases (0°and 180°) of the carrier to transmit binary +1 and -1. Thus BPSK signalspace geometry is one dimensional. QPSK, on the other hand, can beconsidered as two BPSK systems in parallel; one with a sine carrier, theother with a cosine carrier of the same frequency. QPSK signal space isthus two dimensional. This increase in dimension without altering thetransmission bandwidth increases the bandwidth efficiency by a factor oftwo. Spectral compactness is further enhanced in MSK by using acosinusoidal data shaping pulse instead of the rectangular one of QPSK.Though MSK and QPSK use different data shaping pulses, their signalspace geometries are the same. Both of them use a set of fourbiorthogonal signals. So the spectral compactness achieved in MSK overQPSK should be distinguished from the compactness achieved in QPSK overBPSK. In the former compactness comes from the shaping of the datapulse, while in the later it comes from increasing the dimension withinthe given transmission bandwidth.

To see the possibility of any further increase in dimension withoutincreasing the transmission bandwidth substantially, one has to lookinto the time-bandwidth product. It is a mathematical truth that thespace of signals essentially limited in time to an interval τ and in onesided bandwidth occupancy to W is essentially 2τW-dimensional. Thoughthis bound on dimension is true for the best choice of orthonormal set(prolate spheroidal wave functions |4|), yet it will justify thereasoning behind any search for higher dimensional signal sets toachieve higher bandwidth efficiency. In both QPSK and MSK, signalduration τ is 2T, where T is the bit interval in the incoming datastream. Suppose the channel is bandlimited to 1/2T on either side of thecarrier, i.e. one sided bandwidth occupancy is W=1/T. With such abandlimited channel a QPSK system will be able to transmit only ninetypercent of its total power while an MSK system transmits ninety sevenpercent. The number of dimensions available within this bandwidth W=1/Tis 2τW=4. It is surprising that only two of them are utilized in QPSKand MSK. The remaining two are yet left to be played around with. So onecan conceive of a modulation scheme with a bandwidth efficiency as muchas twice that of QPSK or MSK. Since prolate spheroidal wave functionsare not easy to implement, expectation of one hundred percent increasein bandwidth efficiency may be too much from a practical view point. Yetthe extra two dimensions give some room for improving the bandwidthefficiency by increasing the dimensionality of the signal set at thecost of some extra bandwidth, if necessary.

First we briefly review some existing modulation schemes such as QPSK,OQPSK and MSK, all of which use two dimensional signal sets. Then wepropose a new modulation scheme which uses the vertices of a hyper cubeof dimension four as the signal space geometry. This proposed schememakes use of two data shaping pulses and two carriers which are pairwisequadrature in phase; so it is named quadrature-quadrature phase shiftkeying or Q² PSK. It is pointed out as a theorem that in the presence ofAWGN any modulation scheme which utilizes the vertices of some hypercube as signal space geometry has the same energy efficiency; this istrue for any dimension of the hyper cube. As a consequence of thetheorem, Q² PSK has the same energy efficiency as that of MSK, QPSK orOQPSK; but for a given bandwidth the transmission rate of Q² PSK istwice that of any one of the three other schemes. However, all thesefour schemes respond differently when they undergo bandlimiting.Considering ninety nine percent bandwidth of MSK as the definition ofchannel bandwidth, it is shown that E_(b) /N₀ requirement for achievinga BER of 10⁵ is 11.2 db for bandlimited Q² PSK and 9.6 db for MSK. Thusbandlimited Q² PSK achieves twice bandwidth efficiency of MSK only atthe expense of 1.6 db or forty five percent increase in the average bitenergy. Like MSK, Q² PSK has also self synchronization ability.Modulator-demodulator implementation and a scheme for synchronizationwill be described in detail in the last section.

BRIEF REVIEW OF QPSK, OQPSK and MSK

A block diagram of the QPSK modulation scheme is shown in FIG. 1. Theinput binary data stream {a_(k) } arriving at the rate of 1/T isdemultiplexed into two streams a₁ (t) and a₂ (t). The duration of eachbit in the demultiplexed streams is twice the duration of the incomingbit. Streams a₁ (t) and a₂ (t) are multiplied by sine and cosinecarriers and summed up to form the QPSK signal s_(qpsk) (t): ##EQU1##where φ(t), depending on a₁ (t) and a₂ (t), is one of 0°, ±90° and 180°.Thus carrier phase during any 2T interval is one of the four phases. Inthe next 2T interval, if neither of the two bit streams changes sign,the carrier phase remains the same. If one of them changes sign, a phaseshift of ±90° occurs. A change of sign in both streams causes a phaseshift of 180°. Rapid changes in the carrier phase has seriousdeteriorating effects on the signal and the adjacent channel when itundergoes bandlimiting and hardlimiting operations. Those deterioratingeffects are partially eliminated in offset quadrature phase shift keying(OQPSK) where the two bit streams are not allowed to change their signsimultaneously, thus avoiding the possibility of 180° phase shift. Thisis accomplished by skewing or delaying the bit stream a₂ (t) by anamount of time T as illustrated in FIG. 2. By pulse shaping, furtherelimination of changes in the carrier phase is possible. In fact, it istotally eliminated in minimum shift keying (MSK) where a cosinusoidaldata shaping pulse is used in place of the rectangular one of OQPSK, asillustrated in FIG. 2. This pulse shaping in MSK also brings somespectral compactness over OQPSK. The MSK signal can be written as:##EQU2## where b(t)=-a₁ (t)a₂ (t) and φ(t)=0 are π according to a₁ =+1or -1. Thus MSK signals can have one of two instantaneous frequencies,f_(c) ±1/4T. The spacing between the two frequencies is 1/2T. This isthe minimum spacing with which two FSK signals of duration T can beorthogonal; hence the name minimum shift keying (MSK).

It is, therefore, an object of this invention to provide a method fortransmitting data wherein bandwidth efficiency is improved over knowntransmission systems.

It is another object of this invention to provide a digital transmissionsystem wherein all four available dimensions are utilized.

It is a further object of this invention to provide a digital datatransmission system wherein a significant increase in data transmissioncapacity is achieved, for a given bandwidth requirement, at the expenseof only a modest increase in energy per bit.

It is also an object of this invention to provide a transmission systemwherein data transmission capacity can be effectively doubled withoutproducing intolerable intersymbol interference.

It is still another object of this invention to provide amultidimensional digital transmission scheme wherein the transmittedsignal has a constant envelope.

It is a still further object of the invention to provide a data pulseshape whereby intersymbol interference is inhibited in a bandlimitedtransmission environment.

It is additionally an object of this invention to provide amultidimensional digital transmission scheme which can simply andeconomically be adapted to transmit information in various signal spacedimensions.

SUMMARY OF THE INVENTION

The foregoing and other objects are achieved by this invention whichprovides an information transmission method wherein digital data, in theform of a data stream is demultiplexed to form several, illustrativelyfour, demultiplexed data streams. Each of the demultiplexed data streamsis modulated by a respective modulating signal having a carriercomponent and a pulse-shaping component. The pulse-shaping components inthe respective modulating signals have a quadrature phase relationshipand are preferably sinusoidal and cosinusoidal. In accordance with theinvention, each of the data streams is coded to produce a respectivestream of data words, each such word being formed of a predeterminednumber of the data pulses and a parity check bit. Such coding of thetransmitted signal provides a constant amplitude envelope. In a specificillustrative embodiment where each such data word is formed of threedata pulses, a₁, a₂, and a₃, and a parity check bit, a₄, the value ofthe parity check bit is determined in accordance with: ##EQU3## wherebythe transmitted Q² PSK signal has a constant envelope. In someembodiments of the invention, particularly in band limited environments,the pulse shapes are selected to eliminate intersymbol interference.

Preferably the carrier components of the respective modulating signalsare also in a quadrature phase relationship, and therefore, fourdimensions of data transmission are achieved; two for the carrier andtwo for the data pulses in the demultiplexed data streams. Such a fourdimensional transmission system has a doubled data transmission capacityover conventional systems, without requiring any additional bandwidth.The enhanced capacity is achieved at the expense of a modest increase inthe energy per bit.

In accordance with a further aspect of the invention, four streams ofdata pulses are modulated simultaneously by combining a pulse-shapingsignal with first and second carrier signal components. Each of thecarrier signal components has the same frequency as the other, but witha quadrature phase relationship. Such a combination produces first andsecond composite modulation signals, each having first and secondfrequency components. In one embodiment of the invention, the first andsecond frequency components are symmetrical about a base carrierfrequency. The first and second frequency components associated witheach of the composite modulation signals are combined subtractively toproduce a first pair of modulation signals, and additively to produce asecond pair of modulation signals. The four modulation signals arecombined with respective ones of the streams of data pulses, therebyproducing four modulated streams of pulses having predetermined pulseshapes and quadrature pulse phase relationships with respect to eachother.

In accordance with a still further aspect of the invention, a modulatedinformation signal is demodulated by subjecting it to various stages ofnon-linear operation. In one embodiment of the invention, the modulatedinformation signal is squared. The squared modulated information signalis subjected to low pass filtering to extract a frequency-limitedcomponent, and to band pass filtering for extracting a predeterminedfrequency component corresponding to a multiple of the frequency of thecarrier signal. In a preferred embodiment, the frequency component hastwice the frequency of the carrier signal. The extractedfrequency-limited and predetermined frequency components are subjected afurther non-linear operation. In one embodiment, where the furthernon-linear operation is a signal squaring, first and second timingsignals corresponding to the frequency of the information in themodulated information signal, and the frequency of the carrier signal,are extracted therefrom.

BRIEF DESCRIPTION OF THE DRAWINGS:

Comprehension of the invention is facilitated by reading the followingdetailed description in conjunction with the annexed drawings, in which:

FIG. 1 is a function block and partially schematic representation of aquadrature phase shift keying modulator;

FIG. 2 is a timing diagram comparing staggered data pulses in an OQPSKsystem and an MSK system;

FIG. 3 is a function block and partially schematic representation of aQ² PSK modulator constructed in accordance with the invention;

FIG. 4 is a timing diagram showing the waveforms of the variousdemultiplexed signals;

FIG. 5 is a graphical representation of the spectral densities of OQPSK,MSK, and Q² PSK;

FIG. 6 is a graphical representation of the percentage of power capturedas a function of bandwidth for OQPSK, MSK, and Q² PSK;

FIG. 7 is a graphical representation comparing the spectral densities ofMSK, Q² PSKH, and Q² PSKF modulated signals;

FIG. 8 is a graphical representation of the percentage of power capturedas a function of bandwidth for MSK, Q² PSKH, and Q² PSKF modulatedsignals;

FIG. 9 is a graphical representation of the baseband signal spacegeometry of band-limited Q² PSK;

FIG. 10 is a diagrammatic representation of a coding scheme forfour-level MSK;

FIG. 11 is a function block representation of a Q² PSKH modulatorconstructed in accordance with the principles of the invention;

FIG. 12 is a function block representation of a Q² PSK demodulatorconstructed in accordance with the invention;

FIG. 13 is a function block representation of a synchronizationarrangement for demodulating Q² PSK signals;

FIG. 14 is a representation of a plurality of waveforms referenced to acommon time axis;

FIG. 15 is a block and line representation of a constant envelope Q² PSKdemodulator system;

FIG. 16 is a graphical plot of bit error probability P_(b) (E) versusE_(b) /N₀ for MSK and Q² PSK;

FIG. 17 is a block and line representation of a synchronization schemefor coherent demodulation of constant envelope Q² PSK signals;

FIG. 18 is a block and line representation of a baseband model for atransmitter and receiver;

FIGS. 19A, 19B, and 19C are graphical plots of waveforms as respectivetransmitter filter pairs for Q² PSK transmission with zero intersymbolinterference in a bandlimited environment;

FIG. 20 is a graphical plot of bit error probability as a function ofE_(b) /N₀ for Q² PSK transmission using the transmitter filter pairs ofFIGS. 19A, 19B, and 19C;

FIG. 21 is a graphical representation of truncated Sinc function datashaping components for Q² PSK (n=3);

FIGS. 22A, 22B, and 22C are graphical plots comparing the spectraldensities of MSK and Q² PSK using simple, composite, and truncated Sincfunction waveshapes, respectively;

FIG. 23 is a plot of percent power captured as a function of normalizedbandwidth for MSK and Q² PSK (n=3) using Sinc, simple, and compositewaveshaping; and

FIG. 24 is a plot of bit error probability P_(b) (E) as a function ofE_(b) /N₀ for Q² PSK (simple wave shaping), MSK, and QORC.

DETAILED DESCRIPTION

FIG. 3 is a function block representation of a Q² PSK modulatorconstructed in accordance with the principles of the invention. As showntherein, an input data stream a_(k) (t) which is demodulated via aserial to parallel converter into four demultiplexed data streams, a₁(t) to a₄ (t).

QUADRATURE QUADRATURE PHASE SHIFT KEYING

Let us consider the following basis signal set:

    s.sub.1 (t)=cos (πt/2T) cos 2πf.sub.c t, |t|≦T                            (3a)

    s.sub.2 (t)=sin (πt/2T) cos 2πf.sub.c t,|t|≦T                          (3b)

    s.sub.3 (t)=cos (πt/2T) sin 2πf.sub.c t,|t|≦T                          (3c)

    s.sub.4 (t)=sin (πt/2T) sin 2πf.sub.c t,|t|≦T                          (3d)

    s.sub.1 (t)=0,i=1,2,3,4,|t|>T            (3e)

We write,

    p.sub.1 (t)=cos (πt/2T) and                             (4a)

    p.sub.2 (t)=sin (πt/2T)                                 (4b)

Later p₁ (t) and p₂ (t), which are quadrature in phase, will beidentified as data shaping pulses, and sine and cosine functions offrequency f_(c) as carriers. It is to be noted that between any twosignals in the set {s₁ (t)}, there is a common factor which is either adata shaping pulse or a carrier component; the remaining factor in oneis in quadrature with respect to the remaining factor in the other. Thismakes {s₁ (t)} a set of four equal-energy orthogonal signals under therestriction:

    f.sub.c =n/4T,n=integer≧2                           (5)

Also the orthogonality remains invariant under the translation of thetime origin by multiples of 2T, which is the duration of each signal. Inother words, if the definition of s₁ (t) in (3) be extended for all t,then one will get orthogonality over every interval of 2T centeredaround t=2mT, m being an integer.

The orthogonality of {s₁ (t-2mT)} suggests a modulation scheme, aschematic diagram of which is shown in FIG. 3. Data from an IID binary(±1) source at a rate 2/T is demultiplexed into four streams {a₁ (t)};duration of each data pulse (rectangular shaped with strengths ±1) inthe demultiplexed streams being 2T. Each data stream a₁ (t) ismultiplied by the output s₁ (t) of a signal generator which continuouslyemits s₁ (t), defined over all t. The multiplied signals are summed toform the modulated signal s(t).

At the receiver, suppose four identical coherent generators areavailable. Then one can make observations over intervals of length 2Tand use the orthogonality of {s₁ (t-2mT)} to separate out the datastreams. A correlation receiver can perform this process of demodulationin an optimum sense of minimum probability of error in the presence ofwhite Gaussian noise.

The modulating signal s₁ (t) has two fold effect on the bit streams a₁(t): one is the wave shaping of the data pulse; the other is thetranslation of the baseband spectrum to a bandpass region. Shaping ofthe data pulses is illustrated in FIG. 4. It is to be noted that the twopulse trains associated with either carrier are orthogonal over anyinterval of integer multiple of 2T. This makes sense because thedimensionality of the signal set used in this scheme is four; two ofthem come from the orthogonality of the carriers, the remaining two fromthe orthogonality of the data shaping pulses p₁ (t) and p₂ (t). In otherwords, two carriers and two data shaping pulses are pairwise quadraturein phase. Hence the modulation scheme is named Quadrature-QuadraturePhase Shift Keying (Q² PSK).

The bit rate R_(b) =2/T of the input of the modulator in FIG. 3 is twicethe bit rate we considered for QPSK and MSK schemes in the last section.This increase in the rate of transmission is due to increase in thesignal space dimensions and as conjectured earlier, will result in asubstantial increase in the bandwidth efficiency. But for a quantativecomparison of the bandwidth efficiencies of Q² PSK and MSK one needs toknow the spectral occupancy of the Q² PSK signal. This aspect of theinvention will be discussed hereinbelow.

From the schematic diagram in FIG. 3, one can represent the Q² PSKsignal as: ##EQU4## where,

    b.sub.14 (t)=a.sub.1 (t)a.sub.4 (t)                        (7a)

    φ.sub.14 (t)=0 or π according as a.sub.1 (t)=+1 or 1 (7b)

and,

    b.sub.23 (t)=+a.sub.2 (t)a.sub.3 (t)                       (8a)

    φ.sub.23 (t)=0 or π according as a.sub.3 (t)=+1 or -1 (8b)

Thus at any instant the Q² PSK signal can be analyzed as consisting oftwo signals; one is cosinusoidal with frequency either of (f_(c) ±1/4T),the other is sinusoidal with frequency either of (f_(t) ±1/4T). Theseparation between the two frequencies associated with either of the twosignals is 1/2T; this is the minimum spacing that one needs for coherentorthogonality of two FSK signals as in MSK. Also a comparison with (2)shows that the cosinusoidal part of Q² PSK signal in (6b) exactlyrepresents an MSK signal. Therefore the Q² PSK signalling scheme can bethought as consisting of two minimum shift keying type signallingschemes, which, in some loose sense, are in quadrature with respect toeach other. Since the two schemes are in quadrature, one can intuitivelythink that overall energy efficiency will be the same as that ofconventional MSK with coinusoidal shape of data pulse.

FIG. 4 shows the wave shaping of the data pulses in the Q² PSK signal.As shown in this specific illustrative embodiment of the invention, thedata pulses have either sinusoidal or cosinusoidal shapes, therebyproviding a quadrature phase relationship therebetween. However, otherpulse shapes having the quadrature phase relationship may be used in thepractice of the invention.

ENERGY EFFICIENCY

An ultimate objective of all data communication systems is to reduce thebit error rate (BER) at the expense of a minimum amount of average bitenergy (E_(b)). In practice, BER performance is usually evaluated underthe assumption of an ideal channel corrupted only by additive whiteGaussian noise with two sided spectral density N₀ /2. The receiver isassumed to be an optimum one, e.g. a correlation receiver, whichmaximizes the probability of correct decision. A standard quantitativeparameter for measuring BER performance is the energy efficiency (e); itis the ratio E_(b) /N₀ required to achieve a BER P_(b) (E)=10⁻⁵.

The signal set {s₁ (t)} used in Q² PSK is of dimension N=4. Each s₁ (t)represents one of four co-ordinate axes. With respect to this set ofaxes, a Q² PSK signal can be represented as:

    s(t)=[a.sub.1 (t),a.sub.2 (t),a.sub.3 (t),a.sub.4 (t)]     (9)

where the coordinates a₁ (t)'s can have only one of two values ±1 withprobability one half. The number of signals in the Q² PSK signal set is2⁴. The signals are equally probable and of equal energy, say E_(s).Also it is easy to check that they represent the vertices of a hypercube of dimension N=4; the center of the cube being at the origin of thecoordinate axes. For this signal space geometry, the signal errorprobability for any N, |5| is given by:

    P.sub.s (E)=1-(1-p).sup.N                                  (10)

where, ##EQU5## Knowing signal error probability, one has upper andlower bound on bit error probability given by:

    1/4P.sub.s (E)≦P.sub.b (E)≦P.sub.s (E)       (12)

However an exact calculation of P_(b) (E) is of considerable interestfor comparing two modulation schemes. To do that we establish thefollowing theorem.

Theorem:

In the presence of additive white Gaussian noise (AWGN) any modulationscheme which uses the vertices of some hyper cube as signal spacegeometry and an optimum receiver for detection has identical bit errorprobability given by: ##EQU6## where E_(b) is the average bit energy andN₀ /2 is the two sided spectral density of AWGN. This probability oferror holds for any dimension N of the hyper cube. The hyper cube isassumed to be placed symmetrically around the origin to minimize therequirement of average bit energy.

Proof:

Suppose the hyper cube is of dimension N. Then the number of signals inthe modulated signal set is 2^(N) ; each of these signals represents acombination of N bits. If P_(b1) (E) is the probability of error in thei^(th) bit position, then the average bit error probability is: ##EQU7##where the last equality comes form the equality of P_(b1) (E) for all ibecause of the symmetry in signal space geometry. To calculate P_(b1)(E) let us divide the signals into two groups: {(+1,a₂,a3s . . . ,a_(N))} and its image partner {(-1,a₂,a₃ , . . . , a_(N))}, where a₁ 'scan be either of ±1 with probability one half. These two groups ofsignals will lie on two parallel hyper planes of dimension (N 1). Thenone can imagine another hyper plane of the same dimension whichseparates the two groups on its two sides and is equidistant from eachgroup. The distance of any signal in either group from the midway hyperplane is d/2=√E_(s) /N=√E_(b). Thus the signals with +1 in the first bitposition are on one side of this plane at a distance √E_(b) while thesignals with -1 in the first bit position are on the other side at thesame distance. So an error in the first bit position occurs only whenthe noise component n(t) associated with this bit position drives asignal down to the other side of the hyper plane. The probability ofsuch an incident is: ##EQU8## where P_(n) (x) is the probability densityfunction of Gaussian noise with two sided spectral density N₀ /2. Hencethe overall bit error probability is: ##EQU9##

Since we have not assumed any particular value for N, probability oferror given by (16) is valid for any dimension N of the hyper cube;hence the theorem.

From equation (10) one may observe that as N becomes infinitely large,signal error probability P_(s) (E) goes to unity; this is true becauseif signal energy E_(s) is fixed and dimension gets higher and higher,the signals become closer and closer. On the other hand, the abovetheorem asserts a bit error probability P_(b) (E) independent of thedimension N. The explanation of this apparent contradiction lies in thefollowing fact: in the derivation of the theorem we assumed a fixed bitenergy E_(b). So the signal energy E_(s) no longer remains fixed; itincreases linearly with the increase in dimension N. Thus the distancebetween the two hyper planes containing {(+1,a₁,a₂ , . . . , a_(N))} and{(-1,a₁,a₂ , . . . , a_(N))} remains fixed at d=2√E_(b) and therefore,P_(b) (E)=Q(√2E_(b) /N₀) remains fixed while P_(S) (E) does go to unity.In fact the above theorem illustrates that the hyper cube signal spacegeometry coupled with equally probable use of all vertices is equivalentto two-dimensional antipodal geometry.

The bit error probability given by the theorem implies a 9.6 db energyefficiency. BPSK uses two antipodal signals which can be considered asthe vertices of a hyper cube of dimension one. Similarly QPSK and MSK,which use a set of four biorthogonal signals, can be considered as usingthe vertices of a hyper cube of dimension two. And Q² PSK uses thevertices of a hyper cube of dimension four. So all of BPSK, QPSK, OQPSK,MSK and Q² PSK belong to the same class of signalling schemes which usevertices of some hyper cube, and each of them has an energy efficiency9.6 db; this is true when the channel is wide band and corrupted by AWGNonly. If, in addition, the channel is bandlimited, as it happens to bein most practical situations, each of the five schemes respondsdifferently. Due to intersymbol interference signal space geometry nolong remains hyper cube and the energy efficiency is changed. To analyzethe energy in bandlimited situation one needs to know about the spectraldistribution of power and the effect of bandlimiting on signal spacegeometry. We will do those analysis in the next section.

SPECTRAL DENSITY AND EFFECT OF BANDLIMITING

Spectral Density:

One can represent a Q² PSK signal as: ##EQU10## where the additional1/√T is just a normalizing factor to make ##EQU11## unit energy pulses.Data streams a₁ (t)¹ s are independent and at any instant each a₁ (t)can be either +1 or -1 with probability one half. So the Q² PSK signalcan be one of sixteen possible equally probable waveforms. Let usrepresent these waveforms by m₁ (t), i varying from 1 to 16. Probabilityof occurrence of m₁ (t) is p₁ =1/16 for all i. The signal set {m₁ (t)}has the following characteristics:

(i) for each waveform m₁ (t), there is also a waveform -m₁ (t)

(ii) the probability of m₁ (t), and -m₁ (t) are equal

(iii) the transitional probabilities between any two waveforms are thesame.

Such a signalling source is said to be negative equally probable (NEP);the overall spectral density is given by |6|: ##EQU12## where M₁ (f) isthe Fourier transform of m₁ (t). One can reasonably assume the carrierfrequency f₁ >>1/T; then for f>0, each M₁ (f) is one of the followingsixteen possible combinations:

    1/2{±P.sub.1 (f-f.sub.c)±P.sub.2 (f-f.sub.c)±jP.sub.1 (f-f.sub.c)±jP.sub.2 (f-f.sub.c)}

where P₁ (f) and P₂ (f) are the Fourier transforms of data shapingpulses normalizes p_(1n) (t) and p_(2n) (t) given by: ##EQU13##Substituting the M₁ (f)'s into (18) and noticing that all cross termsare cancelled out, one can write:

    S.sub.q.spsb.2.sub.psk (f)=1/2[|P.sub.1 (f-f.sub.c)|.sup.2 +|P.sub.2 (f-f.sub.c)|.sup.2 ]                                                         (21)

The equivalent baseband version of the spectral density is ##EQU14## TheFourier transforms of the data shaping pulses are: ##EQU15##

Substituting (23) and (24) into (22), the spectral density is found tobe: ##EQU16##

Similarly spectral densities of MSK and OQPSK signalling schemes |2| aregiven by: ##EQU17## where in all cases: ##EQU18##

Spectral densities of OQPSK, MSK and Q² PSK are sketched in FIG. 5 as afunction of normalized frequency f/R_(b), where R_(b), the bit rate, is1/T for MSK and 2/T for Q² PSK. It should be noted that for a given bitrate, the width of the main lobe in Q² PSK is just one-half of the widthof the MSK main lobe. Q² PSK uses two different kinds of data pulses:one is p₁ (t) having a cosinusoidal shape as in MSK, the other is p₂ (t)having a sinusoidal shape. The shape of p₁ (t) is smoother than p₂ (t)in the sense that the later has jumps at t=±T; as a result, for large f.the spectral fall-off associated with p₂ (t) is proportional to f⁻²while that with p₁ (t) is as f⁻⁴. The faster fall-off associated withcosinusoidal shape causes lower side lobes in MSK; side lobes in OQPSKand Q² PSK are of the same order in magnitude but relatively higher thanthose of MSK. But just looking at the spectral lobes does not give anyquantitative feelings about the spectral efficiencies; for that we needa measure of spectral compactness.

A measure of spectral compactness is the percent of total power capturedin a specified bandwidth. This is plotted in FIG. 6. For smallbandwidth, the percent power captured in Q² PSK is smaller than that inOQPSK and MSK. Beyond a bandwidth of 1.2/T, the asymptotic behavior ofQPSK and Q² PSK become almost identical because of their same type ofspectral fall-off as f⁻². MSK captures 99.1% of total spectral power ina bandwidth of W=1.2/T. With the same bandwidth power captured in QPSKand Q² PSK are 90.6 and 91.13 percent respectively. Thus MSK seems to bemore spectrally compact than Q² PSK; yet bandwidth efficiency of Q² PSKis higher because its data transmission rate is twice that of MSK. Anexact calculation of bandwidth efficiency depends on the definition ofbandwidth and the effect of bandlimiting on signal space geometries.

But before carrying out the bandlimiting analysis, a few comments on thespectral fall-off of Q² PSK are worth mentioning. In contrast to MSK,the asymptotic spectral fall-off in Q² PSK is as f⁻² ; this is due toabrupt discontinuities in the data pulse p₂ (t) at t=±T. So, in anattempt to achieve higher spectral compactness one may suggest asmoother pulse for p₂ (t). A reasonable suggestion is to replace thehalf sinusoid by a full sinusoid over |t|≦T; this avoids the sharpdiscontinuities at t=±T and results in MSK like asymptotic spectralfall-off as f⁻⁴. But when the transmission band is finite and is belowthe asymptotic region, asymptotic fall-off has little to do with thespectral efficiency; strength of the first few lobes becomes a primaryfactor. So, in spite of faster spectral fall-off in the new Q² PSK, itsspectral compactness ought to be compared with that of Q² PSK with halfsinusoid as p₂ (t). Henceforth whenever we discuss the two Q² PSK casestogether, we denote the half sinusoid case as Q² PSKH and the fullsinusoid case as Q² PSKF.

The baseband spectral density of Q² PSKF-signal is given by ##EQU19##

For the sake of a clear comparison, the spectral densities of MSK and Q²PSKH are once again plotted in FIG. 7 along with the spectral density ofQ² PSKF. The main lobe of Q² PSKF is wider than that of Q² PSKH comparedto Q² PSKH, the side lobes of Q² PSKF are relatively lower in strength.FIG. 8 compares the spectral compactness of Q² PSKH and MSK; it showsthat unless the bandwidth exceeds 1.25/T, the percent power captured byQ² PSKF is less than that with either Q² PSKH and MSK. With a bandwidthof 1.25/T, which is the 99% power bandwidth of MSK, Q² PSKF captures89.90% while Q² PSKH 91.13% of total power. Thus in spite of fasterasymptotic spectral fall-off, Q² PSKF captures almost the sme (in fact alittle less) power as Q² PSKH. But to make a precise statement aboutwhich of the two schemes is more energy efficient, one needs to lookinto the effect of bandlimiting on signal space geometries and theirconsequences on energy efficiencies. We now do the analysis onbandlimiting.

Effect of Bandlimiting:

Consider an existing MSK scheme which allows a bandwidth of 1.2/T sothat almost the entire spectrum (99.1% power) is available at thereceiver. Suppose the MSK modulator is replaced by a Q² PSK modulatorand the modulator output, before transmission, is bandlimited to 1.2/Taround the carrier frequency f_(c). Our object is to compare the energyand the bandwidth efficiencies of the bandlimited Q² PSK with theexisting MSK scheme. We first consider the half sinusoid case (Q² PSKH);the same analysis will also hold for the full sinusoid case (Q² PSKF).

Thus we are assuming the 99% power bandwidth of MSK as the definition ofchannel bandwidth, i.e. W=1.2/T. The bit rate in MSK then being R_(b)^(msk) =1/T, the bandwidth efficiency is b_(msk) =0.83. The bit rate andthe bandwidth efficiency of both Q² PSKH and Q² PSKF are R_(b)^(q).spsp.2^(psk) =2/T and b_(q).spsb.2_(psk) =1.66 respectively. Thusthere is one hundred percent increase in the bandwidth efficiency overMSK without any change in bandwidth; this increase is evidently due toincrease in the dimensionality of the signal space.

With the above definition of channel bandwidth an MSK signal getsthrough almost undistorted, so the energy efficiency is maintained atits ideal value of 9.6 db. A Q² PSKH scheme, on the other hand, whenbandlimited to 1.2/T, allows transmission of only 91.13% of totalspectral power. Thus there is a loss of some spectral components; thisloss causes spread of the baseband data pulses which in turn causesintersymbol interference (ISI). The effect of this ISI can beequivalently considered as changing the signal space geometry. Thefollowing analysis will show that this change in the geometry results inan energy efficiency which is somewhat higher than the ideal value of9.6 db.

In an attempt to find the new signal space geometry, it has been notedthat because of the orthogonality of the two carriers, the spreading ofthe data pulses associated with either carrier does not have any ISIeffect on the signal components associated with the other carrier. Sothe effect of bandlimiting on the geometry of the baseband signal spaceassociated with either carrier can be analyzed separately andindependently of the other. Once the bandlimited baseband signal spacegeometries are known, the overall signal space geometry of thebandlimited Q² PSK signal immediately follows from the product space ofthe individual baseband signal spaces.

Before bandlimiting, the baseband signal space geometries associatedwith both carriers are identical and each of them is biorthogonal. Sinceidentical pair of data pulses are used on either carrier, afterbandlimiting also the baseband geometries will remain identical; but dueto ISI they will no longer remain biorthogonal. After bandlimiting, thebaseband signal associated with either carrier is of the following form:##EQU20##

Where A is an amplitude factor, p_(1b) (t) and P_(2b) (t) are thebandlimited versions of data pulses p₁ (t) and p₂ (t), and a_(l),k 'sbeing either +1 or -1 represent the information bits over the interval(k-1)T<t<(k+1)T.

Squaring both sides of (29) one can write the squared bandlimited signalas: ##EQU21## The expected value of the squared signal is given by:##EQU22## where we used the facts that:

    E{a.sub.1,j a.sub.1,k }=δjk                          (32a)

    E{a.sub.2,j a.sub.2,k }=δjk                          (32b)

    E{a.sub.1,j a.sub.2,k }=0                                  (32c)

where δ_(jk) is the Kronecker delta.

Hence the average energy per transmission of each bit is given by##EQU23##

In the above analysis we assumed that bandlimiting was carried away byan ideal bandpass filter placed symmetrically around the carrier. It maybe useful to be noted that after this filtering, the truncated spectraP_(1b) (f) and P_(2b) (f) of the bandlimited pulses retain their evenand odd symmetry around the carrier frequency f_(c) ; as a result p_(1b)(t) and p_(2b) (t) are not time limited any more, but they maintainmutual orthogonality. In spite of that mutual orthogonality, finding thesignal space geometry of the signal represented by (29) becomesvirtually impossible unless we observe the following facts.

(i) With a bandwidth of 1.2/T, the data pulse p₁ (t) gets through almostundistorted; 99.1% of its total spectral power lies within thatbandwidth. Therefore ##EQU24##

(ii) With a bandwidth of 1.2/T, the data pulse p₂ (t) gets through withonly 83.17% of its total spectral power, of this 83.17% power, 82.5% isconcentrated over |t|≦3T. Thus the bandlimited pulse p_(2b) (t)concentrates most of its energy (99.2%) over |t|≦3T. So one canreasonably assume that p_(2b) (t) is essentially of duration (-3T,3T).

With the two observations mentioned above (29) reduces to ##EQU25##

In order to find the signal space geometry with respect to a basis set##EQU26## it is sufficient to look at ##EQU27## In the absence of thetwo ISI terms on the right hand side of (35), the signal pointcoordinates would have been one of the four possibilities: A|a₁₀,a₂₀R(0)|, where R(0) is the time cross-correlation between p_(2b) (t) andp₂ (t). But because of ISI, the signal points will also depend on theinformation bits a₂,-1 and a₂,1 which are immediate left and right tothe observation bits a₁,0 and a₂,0. Thus signal space geometry isdependent on the data sequence; depending on the values of a₂,-1 anda₂,1, there are four possible geometries each with a probability of 1/4.

Case I.

    a.sub.2,-1 =+1,a.sub.2,1 =+1

With respect to the basis set ##EQU28## the signal points arerepresented by the following set

    S.sub.+1,+1 =(x.sub.1,y.sub.1)                             (36a)

    S.sub.-1,+1 =(-x.sub.1,y.sub.1)                            (36b)

    S.sub.-1,-1=(-x.sub.1,-y.sub.2)                            (36c)

    S.sub.+1,-1 =(x.sub.1,-y.sub.2)                            (36d)

Where the subscripts on signal points S's represent the values of theobservation bits α₁₀ and α₂₀ and the coordinate values are given by##EQU29##

The signal space geometry has been depicted in FIG. 9. It is to be notedthat the geometry is a rectangular one with unequal sides. The signalpoints corresponding to other combinations of a₂,-1 and a₂,1 aresummarized in the following table.

                  TABLE I                                                         ______________________________________                                        Case     I        II        III     IV                                        α.sub.2,-1 ;α.sub.2,1                                                      +1,+1    -1,-1     +1,-1   -1,+1                                     ______________________________________                                        S.sub.+1,+1                                                                            (x.sub.1,y.sub.1)                                                                      (x.sub.1,y.sub.2)                                                                       (x.sub.2,y.sub.3)                                                                     (x.sub.3,y.sub.3)                         S.sub.-1,+1                                                                            (-x.sub.1,y.sub.1)                                                                     (-x.sub.1,y.sub.2)                                                                      (-x.sub.3,y.sub.3)                                                                    (-x.sub.2,y.sub.3)                        S.sub.-1,-1                                                                            (x.sub.1,-y.sub.2)                                                                     (-x.sub.1,-y.sub.1)                                                                     (-x.sub.3,-y.sub.3)                                                                   (-x.sub.2,-y.sub.3)                       S.sub.+1,-1                                                                            (x.sub.1,-y.sub.2)                                                                     (x.sub.1,-y.sub.1)                                                                      (x.sub.2,-y.sub.3)                                                                    (x.sub.3,-y.sub.3)                        ______________________________________                                    

where x₁,y₁,y₂ values are given in (37) and x₂ ≈0.99 A, x₃ ≈1.01 A, y₃≈0.83 A. The effect of bandlimiting on baseband signal space is thus tochange the square (conventional biorthogonal) geometry into rectangularone. Now, if the two baseband signal spaces associated with two carriersare combined to form the product space, the original hyper cube geometryturns into a rectangular hyper parallelopiped of dimension four. Thehyper parallelopiped is not symmetrically palaced around the origin. Itis important and interesting to note that the hyper parallelopiped sidesare very much data dependent. Essentially there are sixteen differentdata dependent situations, each of which has equal probability ofoccurrence, but different parallelopiped for the signal space geometry.Since geometry is dependent on data sequence, a single particularreceiver cannot be optimum in all situations. So one needs to look for areceiver which minimizes the overall bit error probability; in otherwords the receiver design should not be biased to any particular one ofthe sixteen different geometries.

We consider matched filtering followed by a binary decision on each ofthe four signal axes as a candidate for the receiver. In other words, wecorrelate the received signal with each of the four basis signals {s₁(t)} given by (3) and take a binary decision on each of the fourcorrelator outputs. As we see from Table 1, the binary levels (e.g. x₃=1.01 A, -x₂ =-0.99 A in case IV) at the correlator output are not equalin magnitude. So the optimum threshold which minimizes the probabilityof error lies at the midway of the two levels (e.g. 1/2(x₃ -x₂)) and isdifferent from zero level. This optimum threshold is a function of thedata sequence and therefore is not tractable. In this situation, itshould be observed that Table 1 reflects a particular harmony in theclustering of signal points; the coordinate levels x₁ (or y₁) are notall the same, yet occurrence of the positive level x₁ (or y₁) alwaysaccompanies, with equal probability, a negative level -x₁ (or y₁).Therefore if we always set the binary threshold at zero level, thereceiver will not favour any particular level in any biased way. Withthis setting of threshold the average bit error probability is given by,##EQU30## where ##EQU31## and the function Q(.) has been defined in(11). It follows from (38) and (39) that the E_(b) /N₀ requirement in abandlimited Q² PSKH for a bit error rate of 10⁻⁵ is 11.2 db while thatfor MSK is 9.6 db. The Q² PSKH achieves twice the bandwidth efficiencyof MSK only at the expense of 45% increase in the average bit energy. Arigorous simulation of the Q² PSKH scheme confirmed this result towithin 1.4% error, which probably resulted from the assumption that99.1% of the spectral power lies within the bandwidth of 1.2/T. Incomparison to Q² PSKH, the deteriorating effect of ISI is more severe inQ² PSKF. Bit error probability given by (38) and (39) also holds for Q²PSKF except that the signal point coordinate values are different. Q²PSKF coordinate values are x₁ =1.0 A, x₂ =0.95 A, x₃ =1.05 A; y₁ =0.99A, y₂ =0.64 A, y₃ =0.81 A; This leads to an energy efficiency E_(b) /N₀=12.0 db. Thus Q² PSKF achieves twice the bandwidth efficiency of MSK atthe expense of 73% increase in the average bit energy. We see,therefore, that although Q² PSKF has faster asymptotic spectral falloff, Q² PSKH is superior to Q² PSKF in energy efficiency for the givendefinition of bandwidth as W=1.2/T. However, in an attempt to reduce theenergy efficiency, any increase in bandwidth beyond 1.2/T does not helpeither of the two Q² PSK schemes much unless a substantial loss in thebandwidth efficiency is suffered. So, between two Q² PSK schemes. Q²PSKH is the better one to MSK to increase the bandwidth efficiency by afactor of two over MSK.

To achieve twice the bandwidth efficiency of MSK, Q² PSKH costs about a45% increase in the average bit energy. One may like to compare thisincrease with the increase in bit energy required for a multilevel MSKhaving the same bandwidth efficiency as bandlimited Q² PSKH. The set offour signals used in ordinary MSK is biorthogonal (square geometry); thedata pulse associated with each of the two carriers is either a positiveor a negative cosine pulse of duration 2T, i.e. the possible numbers oflevels in the basic data pulse is two. For a multilevel MSK, in order toachieve twice the bandwidth efficiency of biorthogonal MSK, the numberof amplitude levels in the data pulse must be four. In the next sectionwe do the analysis for energy efficiency of four-level MSK and comparethis scheme with bandlimited Q² PSKH.

The 99% power bandwidth (W) of Q² PSKF is 1.75/T, where 2/T is the bitrate (R_(b)). With this as the definition of channel bandwidth, therewill be no noticeable intersymbol interference (ISI) at the receiver,and therefore, energy efficiency will be retained at 9.6 db, as is thecase of MSK. The bandwidth efficiency will be R_(b) /W=1.14 which is 37%higher than the value 0.83 MSK. Q² PSKF is a constant phase frequencyshift keying (CPFSK) which increases the bandwidth efficiency by 37%over MSK without any requirement of increase in the average bit energy.

Q² PSK vs MULTILEVEL MSK

The four level MSK scheme is similar to the conventional biorthogonalMSK scheme except the fact that here each pair of input data bits (eachbit being of duration T/2) is first coded by a 2:1 coder into one of thefour possible levels L₁, i=1,2,3,4. The stream of coded levels (eachlevel being of duration T) is then treated as the input to aconventional MSK modulator. So, the amplitude of the cosine shaped datapulses, as described in FIG. 2, instead of being only +1 or -1, takesone of the four values from the set {L₁ }⁴ as shown in FIG. 10. Thisfour level scheme therefore accepts twice as many input data bits areordinary MSK. To minimize the average bit energy requirement for a givenprobability of error, the amplitude levels are assumed to be placedsymmetrically around origin as illustrated in FIG. 11. The optimumdecision regions D₁ for each level L₁ are also shown. The coding hasbeen performed in such a way that adjacent levels differ by one bitonly; this will reduce the average bit error rate. If we maintain the99% power bandwidth (W=1.2/T) as the channel bandwidth, there will be nonoticeable intersymbol interference. In that situation, the channel iscompletely defined by a set of probabilities {p_(i),j }⁴ where p_(ij) isthe probability that level L₁ is transmitted and L_(j) is detected. Theaverage bit error probability is given by, ##EQU32## where P_(b1) is thebit error probability if only the i^(th) level were allowed to betransmitted. By trivial reasoning one can write

    P.sub.b1 =1/2(p.sub.12 +2p.sub.13 +p.sub.14)               (41a)

    P.sub.b2 =1/2(p.sub.21 +p.sub.23 +2p.sub.24)               (41b)

and by symmetry,

    P.sub.b4 =P.sub.b1                                         (42a)

    P.sub.b3 =P.sub.b2                                         (42b)

    p.sub.12 =p.sub.23                                         (42c)

Hence the average bit error probability is

    P.sub.b (E)=1/4[2p.sub.12 +2p.sub.13 +2p.sub.24 +p.sub.14 +p.sub.21 ](43)

Now referring to FIG. 11 and writing n(t) for the flat noise component,##EQU33## where, ##EQU34## and

    E.sub.b =5d.sup.2 /8

is the average bit energy.

Similarly,

    p.sub.13 =Q(3r)-Q(5r)                                      (44b)

    p.sub.14 =Q(5r)                                            (44c)

    p.sub.21 =Q(r)                                             (44d)

    p.sub.24 =Q(3r)                                            (44e)

Hence the average bit error probability is

    P.sub.b (E)=1/4[3Q(r)+2Q(3r)-Q(5r)]                        (45)

It follows from (45) that for a bit error rate of 10⁻⁶, a four-level MSKrequires 13.4 db E_(b) /N₀. Thus in achieving twice bandwidth efficiencyof biorthogonal MSK, the four-level MSK requires about 142% increase inthe average bit energy; whereas with Q² PSKH and Q² PSKF the incrementsare 45% and 73% respectively. Thus Q² PSKH turns out to be a more energyefficient candidate to increase the bandwidth efficiency by a factor oftwo over ordinary or biorthogonal MSK. Henceforth whenever we mention Q₂PSK we mean Q² PSKH. Results of this section are summarized in thefollowing table.

                  TABLE II                                                        ______________________________________                                        Signal duration = 2T                                                          Bandwidth = 1.2/T                                                             Type of   MSK                        MSK                                      modulation                                                                              biorthogonal                                                                             Q.sup.2 PSKH                                                                           Q.sup.2 PSKF                                                                         Four-level                               ______________________________________                                        Bandwidth 0.83       1.66     1.66   1.66                                     efficiency                                                                    E.sub.b /N.sub.0 for                                                                    9.6 db     11.2 db  12.0 db                                                                              13.4 db                                  P.sub.b (E) = 10.sup.-5                                                       ______________________________________                                    

Q² PSK MODULATOR DEMODULATOR AND SYNCHRONIZATION SCHEME

A block diagram of a Q² PSK (or Q² PSKH) modulator is shown in FIG. 11.Two phase coherent sine and cosine carriers are multiplied by anexternal clock signal at one eighth the bit rate to produce phasecoherent sine and cosine signals of frequencies ##EQU35## These signalsare then separated by means of narrow bandpass filters and combined withappropriate polarity to form the basis signal set {S_(i) (t)}⁴ ofequation (3). The advantage of deriving the basis signals in thisfashion (instead of generating them independently) is that the signalcoherence and the deviation ratio are largely unaffected by any smallvariation in the incoming data rate. These basis signals are multipliedby the demultiplexed data streams and then added to form the Q² PSKsignal defined in eq.(6).

A block diagram of the Q² PSK demodulator is shown in FIG. 12. Thereceived signal (which is given by eq.(6) in the absence of noise andISI) is multiplied by each of the basis signals individually andintegrated over an interval of 2T. This multiplier-integratorcombination constitutes correlation detection or matched filtering, anoptimum coherent receiver in absence of ISI. Binary decisions followedby integrators give an estimate of the four binary data streams a₁ (t),i=1,2,3,4.

One of the basic problems in coherent demodulation is the recovery ofthe modulating signal phase and bit timing information from the receivedsignal. In the present situation, we need to recover the basis signalset {s₁ (t)} and a clock signal at one fourth the bit rate. Thesesignals can be derived from the Q² PSK modulated signal by a nonlinearoperation, such as squaring, and appropriate filtering as shown in FIG.13.

If the Q² PSK modulated signal (eq.6) passes through a squaring device,at the output we get, ##EQU36## where, ##STR1##

There are five components on the right of (46) which carry the requiredclocking and carrier phase information. But it can be shown that theexpected value of each of these five components vanishes separately. Soto recover the clocking and the carrier phase information, we needfiltering and further nonlinear operation. By a lowpass and a bandpassfiltering of the squared signal one may construct two signals x₁ (t) andx₂ (t) as ##EQU37##

After squaring x₁ (t), x₂ (t) and taking the expectation one can show##EQU38##

Thus, on the average, x₁ ² (t) and x₂ ² (t) contains spectral lines at1/T and 4f_(c). One can use these lines to lock phase-locked loops (notshown) and carry out frequency dividions to recover the clocking and thecarrier information as ##EQU39## and

    x.sub.c (t)=Cos 2πf.sub.c t

Signal x_(cl) (t) provides timing information at a rate of one fourththe bit rate; this timing information is essential for sampling theintegrator output in the demodulator (see FIG. 12). The basis signal set{s₁ (t)} required in the process of demodulation can be constructedeasily by simple manipulation of the signals x_(cl) (t) and x_(c) (t).Recovery of x_(cl) (t) and x_(c) (t) from the received signals₄.spsb._(psh) (t) has been illustrated in block diagram in FIG. 13.

CONSTANT ENVELOPE Q² PSK

One can express the Q² PSK signal as follows:

    s.sub.z.spsb.2.sub.psk =A(t) cos (2πf.sub.c t+θ(t)) (53)

where θ(t) is the carrier phase and A(t) is the carrier amplitude givenby: ##EQU40##

Without any additional constraint, the envelope of the Q² PSK signal isnot constant; it varies with time. FIG. 14 is a representation of aplurality of waveforms referenced to a common time scale, andillustrates that Q² PSK has an envelope which varies in amplitude withtime.

Although energy and bandwidth efficiencies are the two most significantcriterion in the design of a modulation scheme. It is also verydesirable to achieve a constant envelope as a feature of the system,particularly in certain nonlinear types of channels. For example, thetravelling wave tube (TWT) amplifier of a satellite repeater usuallyconverts amplitude variations into spurious phase modulation. A constantenvelope of the transmitted signal would greatly reduce this problem.

In order to maintain a constant envelope let us consider a simple blockcodding at the input of the Q² PSK modulator: the coder accepts serialinput binary data and for every three information bits {a₁,a₂,a₃ }, itgenerates a codeword {a₁,a₂,a₃,a₄ } such that the first three bits inthe code word are the information bits and the fourth one is an oddparity check for the information bits. The rate of the code is 3/4. Onecan write the parity check bit a₄ (t) as ##EQU41##

Substituting (55) into (54) one may observe that if this coded bitstream be modulated by Q² PSK format, a constant envelope is maintained.This additional envelope feature is achieved at the sacrifice ofbandwidth efficiency; the information tramsmission rate is reduced fromR_(b) =2/T to R_(b) =3/2T.

Four of the eight possible code words {C₁ }⁸ ₁₌₁ are follows:

    C.sub.1 =(+++-)

    C.sub.2 =(++-+)

    C.sub.3 =(+-++)

    C.sub.4 =(+---)

The remaining four code words are just the negatives of these. This is aset of eight biorthogonal codes with a minimum distance d_(min) =2. Thecode therefore can not be used for error correction. The redundantinformation associated with the fourth demultiplexed data stream a₄ (t)can be used to improve the signal to noise performance of the code.

In practical situation when the signal is bandlimited, the biorthogonalstructure of the code is destroyed due to intersymbol interference. InQ² PSK format there are two data shaping pulses: one is a smoothcosinusoid associated with data streams a₁ (t) and a₃ (t); the other isa half sinusoid with a₂ (t) and a₄ (t). On bandlimiting, the halfsinusoid, because of its sharp discontinuities at the ends, getsrelatively distorted. So we assume that, at the receiver, the redundantinformation associated with a₄ (t) is used only in making the binarydecision about the information in a₂ (t); the decision about theinformation bits in a₁ (t) and a₃ (t) are made independently from theobservations on the respective pulse trains only. A block diagram of thedemodulator is shown in FIG. 15.

To make a decision about a₂, we make a simplifying assumption that a₁and a₃ are decoded correctly. Correctly made decisions of a₁ and a₃along with the estimates a_(2r) of a₂ and its redundant version a_(4r)of a₄ are then the observations for making decision about a₂. It can beshown that a sufficient statistic for making this decision is the randomvariable V, ##EQU42##

A decoder, which is optimum in the sense of minimizing the probabilityof error, will take a decision a₂ as +1 or -1 according as V≧0 or V<0.In actual situation, however, formation of the right decision statistic(V) is subject to the correctness of the decision about a₁ and a₃. Letp₁ and p₃ be the probability of error in making decision about a₁ and a₃and q be the probability of error in decision of a₂ when decision isbased on correct decision statistic (V). Then one can show that theactual probability of error in a₂ is given by

    p.sub.2 =q+p.sub.1 (1-p.sub.1)(1-2q)                       (57)

In evaluating the performance of the scheme, bandlimiting has beenallowed at both receiver and transmitter through the use of sixth orderButterworth filters with half power bandwidth equal to 1.2/T=0.8 R_(b).It has been found that for a bit error rate of 10⁻⁵, the constantenvelope scheme requires an E_(b) /N₀ =10.3 dB while that for MSK isE_(b) /N₀ =9.5 dB. Thus there is 50% increase in the bandwidthefficiency over MSK at the cost of 0.8 dB increase in the average bitenergy though both schemes have constant envelopes. Bit errorprobability of both MSK and the coded Q² PSK have been plotted againstE_(b) /N₀ in FIG. 16.

The above evaluation of performance is based on the assumption of bit bybit detection. With symbol by symbol detection, one may utilize thedistances among signals of coded Q² PSK (which are biorthogonal) moreefficiently and make considerable improvement in energy efficiency. Inabsence of bandlimiting, with symbol by symbol detection the E_(b) /N₀requirement of coded Q² PSK for a bit error rate of 10⁻⁵ is about 5.5dB.

DEMODULATOR AND SYNCHRONIZATION SCHEME

A block diagram of the receiver which performs coherent demodulation ofthe coded Q² PSK signal is shown in FIG. 15. One of the basic problemsin coherent demodulation is the recovery of the modulating signal phaseand bit timing information from the received signal. In the presentsituation, we need to recover the basis signal set {s₁ (t)} and a clocksignal at a rate of 1/2T. These signals can be derived from the Q² PSKmodulated signal by a nonlinear operation, such as squaring, andappropriate filtering in FIG. 17.

If the Q² PSK modulated signal defined by (53) passes through a squaringdevice, at the output we get, ##EQU43## where, Substituting a₄ =-a₁ a₂/a₃, one can simplify (58) as ##EQU44##

There are two components on the right of (60) which carry the requiredclocking and carrier phase information. It can be shown that theexpected value of each of these two components is zero. So the recoveryof the clocking and the carrier phase information requires filtering andfurther nonlinear operation. By bandpass filtering of the squared signalone may construct two signals x₁ (t) and x₂ (t) as ##EQU45##

After sqaring x₁ (t), x₂ (t) and taking the expectation one can show##EQU46##

Thus x₁ ² (t) and x₂ ² (t) contains spectral lines at 4f_(c) ±1/T. Onecan use these lines to lock phase-locked loops, and carry out frequencydivisions so as to form the signals x₃ (t) and x₄ (t) as ##EQU47## Amultiplication of these two signals followed by bandpass filtering andfrequency division gives clocking and carrier phase information as##EQU48## and

    x.sub.c (t)=cos 2πf.sub.c t                             (68)

Signal x_(cl) (t) provides timing information at a rate of 1/2T; thistiming information is essential for sampling the integrator output inthe demodulator (see FIG. 15). The basis signal set {s₁ (t)} required inthe process of demodulation can be constructed easily by simplemanipulation of the signals x_(cl) (t) and x_(c) (t).

III. Pulse Shaping

Original Q² PSK used a half cosinusoid and a half sinusoid [defined asp₁ (t) and p₂ (t) in 2(a) and 2(b)] as two data shaping pulses. It isobserved that in achieving twice the bandwidth efficiency of minimumshift keying, the original Q² PSK requires about forty five percentincrease in the average bit energy which is mostly due to intersymbolinterference caused by the bandlimiting effect on the data shaping pulsep₂ (t). With the ninety nine percent power bandwidth of MSK as thedefinition of channel bandwidth, a data pulse of the shape p₁ (t) getsthrough almost undistorted; but a data pulse of the shape p₂ (t) isseriously distorted because of the sharp discontinuities in p₂ (t) att=±T. It is also observed that if three or more pulses of the shape p₂(t) occur in a row with the same polarity then bandlimiting causes adestructive interference of worst kind. This worst case situation ofinterference is primarily responsible for the requirement of higher bitenergy. In an attempt to improve the energy efficiency, one way toreduce the intersymbol interference is to apply proper waveshaping tothe data pulses. In other words, one needs to find out suitable shapesp₁ (t) and p₂ (t) so that their effect of intersymbol interference dueto bandlimiting is either eliminated completely or reduced greatly.

In this section we look into Q² PSK transmission in baseband domain.Baseband model of transmitter and receiver is illustrated in FIG. 18. Inactual situation, however, two such blocks of transmitter and receiverare present; one is associated with the sine carrier, the other is withthe cosine carrier. In this model the input binary data stream has beenrepresented by a series of impules occurring at intervals of T sec. Theamplitude factor a_(k) can be either +1 or -1. The input stream isdemultiplexed into two streams; the rate of impulses in thedemultiplexed streams is half of the rate (1/T) in the incoming stream.P₁ (f) and P₂ (f) are the pair of transmitter filters; P₁ *(f) and P₂*(f) are the corresponding matched filters at the receiver. The receiverand transmitter filters occupy a common bandwidth (-W, W), where W isthe one sided bandwidth. In absence of noise n(t), the signal at theinput of receiver is given by, ##EQU49## where p₁ (t) and p₂ (t) are theimpulse responses of the transmitter filters which are not necessarily(unlike original Q² PSK data pulses) time limited and a_(l),k '8 beingeither +1 or -1 represent the demultiplexed information bits over theinterval (k-1)T<t<(k+1)T.

Suppose the two observation bits are a₁,0 and a₂,0. At the receiver thematched filter outputs are sampled at regular intervals of 2T and binarydecisions are taken using a zero crossing detector. The sampled valuesat the output of the two matched filters represent the co-ordinates ofthe baseband signal in a two dimensional signal space where p₁ (t) andp₂ (t) serve as the bases. The coordinate values are given by ##EQU50##where, ##EQU51## are the sampled values of the auto andcross-correlation function of the pulses p₁ (t) and p₂ (t). The firstterm on the right of (70) or (71) is the desired one; the remaining twosummation terms are due to intersymbol interference. It is to be notedthat p₁ (t) and p₂ (t), in general, are not timelimited; so both streamof pulses are causing intersymbol interference. To eliminate the effectof ISI in the process of making binary decisions one needs the followingcriterion

    R.sub.11 (n)=R.sub.22 (n)=δ.sub.no                   (76a)

    R.sub.12 (n)=R.sub.21 (n)=0                                (76b)

where, ##EQU52##

It should be observed that the above set of criterion imposes muchstronger restrictions on the shape of the two data pulses than meresimple orthogonality between them. The auto and cross-correlationfunctions are related to the power spectra by the following relations.##EQU53## where,

    i,j=1,2;i≠j

Now, R_(ll) (k) can be written in the following form ##EQU54## A changeof variables and an interchange of summation and integration gives##EQU55## where, ##EQU56## is the equivalent wrapped version of thebandlimited power spectrum of the i^(th) pulse p_(l) (t). Similarly eq.(76b) can be written as ##EQU57## where, ##EQU58## is the equivalentwrapped version of the bandlimited cross power spectrum of the pulsesp_(i) (t) and p_(j) (t). The equivalent wrapped version is constructedby slicing the original spectrum into segments of width 1/2T andsuperimposing all the segments on the interval [-1/4T, 1/4T].

In an attempt to improve the energy efficiency of Q² PSK, we, therefore,frame the following guidelines to study the scope of waveshaping inreducing the effect of ISI:

Chosse P_(l) (f),i=1,2 such that ##EQU59## are satisfied for completeelimination of ISI or the above two integrals are minimized so that theeffect of ISI is minimized in the sense of minimizing the probability oferror.

There are infinite number of solutions for P₁ (f) and P₂ (f) whichsatisfy (84a) (84b). A few of them are illustrated in FIGS. 19A, 19B,and 19C. From the realization point of view the solution illustrated inFIG. 19C is specially convenient; one of the two filters has a raisedcosine shape P₁ (f) with 100% roll-off (i.e. 100% excess bandwidth), theother is P₂ (f) which is just a bandlimited differentiator. With thispair of filter shapes, the bandwidth efficiency is 2 bits/sec/hertzwhich is considerably higher than the bandwidth efficiency 1.66 oforiginal Q² PSK. The energy efficiency is also improved from originalvalue of 11.2 dB to 9.6 dB in present situation. But it should be notedthat P₂ (f) has a realization problem due to the sharp cut off atf=±1/2T.

However, instead of sharp cut off filtering at f=±1/2T if one allows aButterworth filtering alongwith an ideal differentiator, the realizationproblem is greatly reduced. FIG. 20 shows the bit error probabilityperformance with P₁ (f)=√2T cos πfT and P₂ (f)=fB(f), where B(f) is aButterworth filter of second order with three dB bandwidth as W=0.5R_(b). The corresponding bandwidth efficiency is 2 bits/sec/Hertz andthe bit energy requirement for a bit error rate of 10⁻⁵ is 10.8 dB. Thuspulse shaping provides considerable improvement in both energy andbandwidth efficiencies over the original Q² PSK.

IV. Generalized Quadrature-Quadrature Phase Shift Keying

Let {p₁ (t)}^(n) _(l=1) be a set of n data shaping components, which,along with two orthogonal carriers of frequency f_(c), are used tomodulate 2n IID binary data streams {a,(t)}^(n) _(l=1) and {b_(l)(t)}^(n) _(l=1) such that the modulated signal is given by, ##EQU60##where, ##EQU61##

Each binary data (±1) in a₁ (t) and b₁ (t) is of duration T; e.g. the kth data in any of the 2n streams appears over (2k-1)T/2 to (2k+1)T/2.The modulated signal space will be the vertices of a hypercube ofdimension 2n. The overall bit rate of the system is R_(b) =2n/T. Signals(t) represents a generalized Q² PSK signal. The special case of n=1 isa two dimensional scheme such as QPSK,MSK or QORC [Appendix A]. In thefirst section we considered the special cases of n=2 which is theoriginal Q² PSK

Phase and spectral properties of Q² PSK(n=2)

One can write Q² PSK(n=2) signal as

    s.sub.q.spsb.2.sub.psk (t)=A(t) cos [2πf.sub.c t+θ(t)](87)

where, ##EQU62##

Symbol transitions occur at the instants t=(2n+1)T/2, where n is aninteger; at those instants carrier phase θ(t) can be any one of the fourpossible values ±45°, ±135°. Thus an abrupt ±90° or 180° phase change inthe RF carrier will occur at every symbol transition instant. Thus themodulated signal doesn't maintain continuity in phase. In designing amodulation scheme, though energy and bandwidth efficiencies are the twomost important criterion, continuity of phase in the RF signal may be anadditional desirable feature in certain situations. With continuity inphase, high frequency content and therefore secondary sidelobes can beexpected to be relatively lower in strength; in other words, spectralfall-off will be sharper and so restrictions on the subsequentbandlimiting filter shapes can be relaxed. This is desirable in certainssituations where filtering after modulation is cost prohibited and outof band radiation needs to be at low level. Also in bandlimitedsituation, faster spectral fall-off of the signal itself may result inless ISI and hence less average bit energy requirement for specified biterror rate. So it is quite reasonable to look for possibilities ofbringing continuity of phase in the Q² PSK signal. In order to bringphase continuity, we look at the Q² PSK signal in a six dimensionalsignal space.

Q² PSK (n=3)

In general, the Q² PSK signal can be written as

    s(t)=A(t) Cos [2πf.sub.c t+θ(t)]                  (89)

where A(t) is the amplitude of the modulated carrier and θ(t) is thephase given by ##EQU63##

We assume the data shaping components p₁ (t) are continuous for all tand the modulated signal maintains continuity in phase at every symboltransition instant; this is desirable for faster asymptotic fall off inthe spectral density of the modulated signal. This condition issatisified if

    p.sub.1 [(2k+1)T/2]=0 for all k                            (91)

The minimum bandwidth solution for p₁ (t) is a sinc function whichoccupies a bandwidth of |f|≦1/T. But the minimum bandwidth solutionbeing unique, one can't have more than one p₁ (t) which maintainsorthogonality (86) and occupy the same minimum bandwidth |f|≦1/T at thesame time. So we looked for a different (nonoptimum) set of pulseshaping components and compared their spectral efficiencies. One choiceis a truncated sinc function given by ##EQU64## where A=1.0518 is anormalizing constant to make x(t;T) a unit energy pulse. In order tomaintain orthogonality among the data shaping components we constructedthe set {p₁ (t)} as follows:

    p.sub.1 (t)=x(t;T)                                         (93a)

    p.sub.2 (t)=[D.sub.2.sup.1 -D.sub.2.sup.-1 ]x(t;T/2)       (93b)

    p.sub.3 (t)=[-D.sub.3.sup.1 +1-D.sub.3.sup.-1 ]x(t;T/3)    (93c) ##EQU65## where n=2m+1 and the operator D.sub.j.sup.k represents a delay of kT/j units of time. The construction of this set of data shaping components is illustrated in FIG. 21.

Generation of sophisticated data pulse at high frequency (such as atruncated sinc pulse at several GHz) may become very expensive. So otherchoices of data shaping components which are less expensive are worthmentioning. We consider the following set of three signals as an example##EQU66## This set of data shaping components satisfy both orthogonaland zero crossing condition given by (86) and (91). Each of thesesignals can be expressed as sum or difference of two signals offrequencies f₁ =1/2T and f₂ =3/2T, so we call (93) composite waveforms.

The difference between the two frequencies is 1/T; this is exactly twicethe minimum spacing (1/2T) that one needs for coherent orthogonality oftwo FSK signals of duration T. So there must be another set of threesignals with frequences f₁, f₂, f₃ such that the difference betweenadjacent consecutive frequencies is the minimum spacing needed forminimum shift keying. This set is given by ##EQU67## We call (95) simplewave forms. The set constructed from truncated sinc function becomesidentical to this set of simple wave forms if one redefines x(t;T) in(92) as cos (πt/T). The effect of modulating this set of simple datashaping components by orthogonal carriers is to translate the spectrumof a set of minimum shift keying type signals (which is different fromthe well known MSK scheme) from baseband to a bandpass region. This isthe basic difference between quadrature-quadrature phase shift keyingand minimum shift keying which enables Q² PSK to use a given bandwidthmore efficiently in the bandpass region.

Next we will discuss the spectral properties of Q² PSK schemes whichutilize the set of truncated sinc pulses, composite waveforms and simplewaveforms as the baseband data shaping components. Among these threesets of data shaping components, particularly the last two arefavourably from an implementation view point; this is because they couldbe constructed by simple addition and subtraction of sinusoids which canbe generated with perfection even at several GHz frequency.

Spectral Density

The equivalent normalized baseband power spectral density of generalizedQ² PSK signal is given by, ##EQU68## where P₁ (f) is the Fouriertransform of the i^(th) data shaping pulse p₁ (t) and ##EQU69##

Simple Wave Shape

The baseband data shaping components over |t|≦T/2 are given by (95); thecorresponding Fourier transforms are ##EQU70## Hence the spectraldensity is given by ##EQU71## where the additional superscript s standsto mean simple wave shaping.

Composite Wave Shape

The baseband data shaping components over |t|≦T/2 are given by (94); thecorresponding Fourier transforms are given by ##EQU72## Hence thespectral density is given by ##EQU73## where the additional superscriptc stands to mean composite wave shaping.

Truncated Sinc Function Wave Shape

The data shaping components are given by (93); we consider only thefirst three components to form a six dimensional signal space. There isno closed form expression for the spectral density; results due tonumerical calculation are illustrated in FIG. 22 and FIG. 23.

FIG. 22 shows the spectral densities of MSK and three continuous phaseQ² PSK cases, spectral densities have been plotted against normalizedfrequency f/R_(b), where R_(b) is the bit rate associated withrespective schemes. It should be noted that the spectral fall-offassociated with all three generalized Q² PSK schemes, each of whichmaintains continuity in phase, is much faster than that associated withMSK, although MSK also maintains a continuity in the carrier phase.Among three generalized Q² PSK schemes, simple and composite waveshaping exhibit almost identical spectral density (except a few nulls inthe composite case) which fall off somewhat faster than that oftruncated sinc waveshaping. But a mere looking into the spectraldensities doesn't give any quantitative information about spectralcompactness and energy efficiencies of the schemes.

A measure of spectral compactness is the percent of total power capturedin a specified bandwidth; this is plotted in FIG. 23 for all fourschemes, where W represents one sided bandwidth around the carrierfrequency. We consider ninety nine percent power bandwidth as theinverse measure of spectral compactness. FIG. 23 shows that both simpleand composite waveshaping reaches quickly to the ninety nine percentvalue at W=0.75R_(b) and at W=0.725R_(b) respectively; whereas truncatedsinc waveshaping and MSK reaches that value at W=0.9R_(b) andW=1.2R_(b). Therefore in an increasing order of spectral compactness,the four schemes come in the following order: MSK, Q² PSK(n=3):Truncated Sinc Wave Shape, Simple and Composite Wave Shape; thecorresponding bandwidth efficiencies, defined to be the ratio of bitrate to bandwidth, are 0.83, 1.1, 1.33 and 1.38 respectively. Thus withcomposite waveshaping it is possible to achieve a bandwidth efficiencywhich is 1.65 times that of MSK. It should be remembered that Q² PSK(n=2) increases the bandwidth efficiency by a factor of two over MSK. Inapplications where bandwidth efficiency is of primary concern and Q² PSK(n=2) seems to be a better candidate; but where use of fine filtering(bandlimiting) is cost prohibitive or symbol switching is desirable atzero amplitude level, Q² PSK(n=3) may find a suitable position.

If transmission is allowed over 99% power bandwidth of respectivemodulation schemes, one may reasonably assume that there will be nonoticeable intersymbol interference and the hypercube geometry of thesignal will virtually remain unchanged. Hence for Q² PSK(n=3) schemeswith bandwidth equal to 0.725R_(b) and 0.75R_(b) respectively forcomposite and simple waveshaping, the average bit energy requirement(E_(b) /N₀) is 9.6 db. One may summarize all the results in thefollowing table.

                                      TABLE I                                     __________________________________________________________________________    Bit Rate = R.sub.b                                                            Type of MSK    Q.sup.2 PSK(n = 3)                                                                    Q.sup.2 PSK(n = 3)                                                                    Q.sup.2 PSK(n = 3)                                                                    Q.sup.2 PSK                            modulation                                                                            biorthogonal                                                                         trun.sinc                                                                             simple  composite                                                                             n = 2                                  __________________________________________________________________________    Bandwidth                                                                             .83    1.11    1.33    1.38    1.66                                   efficiency                                                                    E.sub.b /N.sub.0 for                                                                  9.6 dB 9.6 dB  9.6 dB  9.6 dB  11.2 dB                                P.sub.b (E) = 10.sup.-5                                                       __________________________________________________________________________

Q² PSK performance has also been compared with that of QORC, another nonconstant envelope scheme. Though spectral density of QORC falls off asf⁻⁶, while that of Q² PSK as f⁻⁴, the latter scheme outperforms theformer in bandwidth efficiency. Bit error probability of Q² PSK(simple), MSK and QORC are plotted in FIG. 24 as function of E_(b) /N₀.This result is based on bandlimiting using sixth order Butterworthfilter with half power bandwidth as: W=R_(b) for both MSK and QORC,W=2/3R_(b) for Q² PSK. Two different receivers have been considered forQORC (ref. to Appendix A): one is MSK, the other is QPSK. It is observedthat the performances of Q² PSK and QORC with the MSK receiver arealmost identical; while energy efficiency (ie E_(b) /N₀ for a P_(b)(E)=10⁻⁵) of MSK is slightly better by 0.4 db. However, Q² PSK transmitsfifty percent more data than QORC and MSK.

Appendix A Two Dimensional Schemes: QPSK, MSK, QORC

A two dimensional modulated signal can be represented by (85) with n=1;it is given by

    s(t)=a.sub.1 (t)p.sub.1 (t) cos 2πf.sub.c t+b.sub.1 (t) p.sub.1 (t) sin 2πf.sub.c t                                            (A1)

For QPSK, ##EQU74## For MSK (except for a relative offset of T/2),##EQU75##

Both QPSK and MSK are constant envelope modulation schemes; theirspectral densities fall off as f⁻² and f⁻⁴ respectively. A third kind oftwo dimensional non constant envelope scheme can be described by a pulseshaping component as ##EQU76## Where A is a constant and * denotes timeconvolution. Here h(t) represents a raised cosine pulse which extendsfrom -T to +T and therefore causes overlapping with the adjacent pulseson either side; hence the name is Quadrature Overlapped Raised Cosine(QORC) scheme. It spectral density is the product of QPSK and MSKspectral densities and therefore fall off as f⁻⁶. The spectral densityis given by ##EQU77## In addition to f⁻⁶ spectral fall off, QORC retainsthe same first null as QPSK (the first null of MSK is at 1.5 times thatof QPSK). QORC is more spectrally compact than MSK and QPSK, but itdoesn't maintain constant envelope. Because of overlapping of theadjacent pulses, matched filter detection is not optimum; detectionswith QPSK or MSK demodulators are two possibilities. In spite of its nonconstant envelope, QORC has been reported to outperform MSK in certainnonlinear channels.

Although the invention has been described in terms of specificembodiments and applications, persons skilled in the art, in light ofthis teaching, can generate additional embodiments without exceeding thescope or departing from the spirit of the claimed invention.Accordingly, it is to be understood that the drawings and descriptionsin this disclosure are proffered to facilitate comprehension of theinvention and should not be construed to limit the scope thereof.

What is claimed is:
 1. A modulation method for modulating simultaneouslyfour streams of data pulses, the modulation method comprising the stepsof:coding each of the four streams of data pulses into a respectivelyassociated stream of data words, each such data word being formed of apredetermined number of the data pulses from the associated one of thedata pulse streams, and a parity check bit, said parity check bit havinga value responsive to a mathematical operation performed using saidpredetermined number of the data pulses; combining a pulse-shapingsignal with respective ones of first and second carrier signalcomponents, each of said first and second carrier signal componentshaving the same frequency as the other and a quadrature phaserelationship with respect to the other, to produce first and secondcomposite modulation signals, each having first and second frequencycomponents; combining subtractively said first and second frequencycomponents associated with each of said first and second compositemodulation signals, to produce first and third modulation signals;combining additively said second and first frequency componentsassociated with each of said first and second composite modulationsignals, to produce second and fourth modulation signals; and combiningsaid first, second, third, and fourth modulation signals with respectiveones of said four streams of said data words, to produce respectivelycorresponding first, second, third, and fourth modulated streams ofpulses having predetermined pulse shapes and quadrature pulse phaserelationships with respect to one another, combining said first, second,third, and fourth modulated streams of pulses, whereby a transmissionsignal formed of the combination of said first, second, third, andfourth modulated streams of pulses has a constant envelope.
 2. Themethod of claim 1 wherein each of said data words is formed of threedata pulses, a₁, a₂, and a₃, and said parity check bit, a₄, where:##EQU78##